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Homogenization, Linearization, and Large‐Scale Regularity for Nonlinear Elliptic Equations
Author(s) -
Armstrong Scott,
Ferguson Samuel J.,
Kuusi Tuomo
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21902
Subject(s) - homogenization (climate) , linearization , mathematics , nonlinear system , lagrangian , mathematical analysis , physics , biodiversity , ecology , quantum mechanics , biology
We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). We also obtain a quantitative estimate on the rate of this homogenization. These results lead to a better understanding of differences of solutions to the nonlinear equation, which is of fundamental importance in quantitative homogenization. In particular, we obtain a large‐scale C 0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large‐scale C 1, 1 estimate for solutions, also with optimal stochastic integrability. Each of these regularity estimates are new even in the periodic setting. As a second consequence of the large‐scale regularity for differences, we improve the smoothness of the homogenized Lagrangian by showing that it has the same regularity as the heterogeneous Lagrangian, up to C 2, 1 . © 2020 Wiley Periodicals LLC